International Association
for Cryptologic Research

Non-interactive zero-knowledge proofs (NIZK) are essential building blocks in threshold cryptosystems like multiparty signatures, distributed key generation, and verifiable secret sharing, allowing parties to prove correct behavior without revealing secrets. Furthermore, universally composable (UC) NIZKs enable seamless composition in the larger cryptosystems. A popular way to construct NIZKs is to compile interactive protocols using the Fiat-Shamir transform. Unfortunately, Fiat-Shamir transformed NIZK requires rewinding the adversary and is not *straight-line extractable*, making it at odds with UC. Using Fischlin's transform gives straight-line extractability, but at the expense of many repetitions of the underlying protocol leading to poor concrete efficiency and difficulty in setting parameters. In this work, we propose a simple new transform that compiles a Sigma protocol for an algebraic relation into a UC-NIZK protocol *without any overheads of repetition*. - Given a Sigma protocol for proving m algebraic statements over n witnesses, we construct a compiler to transform it into a *straight-line extractable* protocol using an additively homomorphic encryption scheme AHE. Our prover executes the Sigma protocol's prover once and computes 2n encryptions. The verification process involves running the Sigma protocol verifier once and then computing n encryptions, which are homomorphically verified against the prover generated encryptions. We apply the Fiat-Shamir transform to the above straight-line extractable Sigma protocol to obtain a UC-NIZK. We instantiate AHE using class group-based encryption where the public key of the encryption scheme is obliviously sampled using a suitable hash function. This yields a UC-NIZK protocol in the random oracle model.

Proof-carrying data (PCD) is a powerful cryptographic primitive that allows mutually distrustful parties to perform distributed computation in an efficiently verifiable manner. Known constructions of PCD are obtained by recursively-composing SNARKs or related primitives. SNARKs with desirable properties such as transparent setup are constructed in the random oracle model. However, using such SNARKs to construct PCD requires heuristically instantiating the oracle and using it in a non-black-box way. Chen, Chiesa and Spooner [Eurocrypt'22] constructed SNARKs in the low-degree random oracle model, circumventing this issue, but instantiating their model in the real world appears difficult. In this paper, we introduce a new model: the arithmetized random oracle model (AROM). We provide a plausible standard-model (software-only) instantiation of the AROM, and we construct PCD in the AROM, given only a standard-model collision-resistant hash function. Furthermore, our PCD construction is for arbitrary-depth compliance predicates. We obtain our PCD construction by showing how to construct SNARKs in the AROM for computations that query the oracle, given an accumulation scheme for oracle queries in the AROM. We then construct such an accumulation scheme for the AROM. To prove the security of cryptographic constructs in the AROM, we give a non-trivial and efficient "lazy sampling" algorithm (a "stateful emulator") for the ARO up to some error. We obtain this construction by developing a toolkit for analyzing cryptographic constructions in the AROM, which uses algebraic query complexity techniques and the combinatorial nullstellensatz.

Succinct non-interactive arguments of knowledge (SNARKs) are cryptographic proofs with strong efficiency properties. Applications of SNARKs often involve proving computations that include the SNARK verifier, a technique called recursive composition. Unfortunately, SNARKs with desirable features such as a transparent (public-coin) setup are known only in the random oracle model (ROM). In applications this oracle must be heuristically instantiated and used in a non-black-box way. In this paper we identify a natural oracle model, the low-degree random oracle model, in which there exist transparent SNARKs for all NP computations *relative to this oracle*. Informally, letting $O$ be a low-degree encoding of a random oracle, and assuming the existence of (standard-model) collision-resistant hash functions, there exist SNARKs relative to $O$ for all languages in $NP^{O}$. Such a SNARK can directly prove a computation about its own verifier. To analyze this model, we introduce a more general framework, the *linear code random oracle model* (LCROM). We show how to obtain SNARKs in the LCROM for computations that query the oracle, given an *accumulation scheme* for oracle queries. Then we construct such an accumulation scheme for the special case of a low degree random oracle.

We present a new multiparty protocol for the distributed generation of biprime RSA moduli, with security against any subset of maliciously colluding parties assuming oblivious transfer and the hardness of factoring. Our protocol is highly modular, and its uppermost layer can be viewed as a template that generalizes the structure of prior works and leads to a simpler security proof. We introduce a combined sampling-and-sieving technique that eliminates both the inherent leakage in the approach of Frederiksen et al. (Crypto’18) and the dependence upon additively homomorphic encryption in the approach of Hazay et al. (JCrypt’19). We combine this technique with an efficient, privacy-free check to detect malicious behavior retroactively when a sampled candidate is not a biprime and thereby overcome covert rejection-sampling attacks and achieve both asymptotic and concrete efficiency improvements over the previous state of the art.

We present a new multiparty protocol for the distributed generation of biprime RSA moduli, with security against any subset of maliciously colluding parties assuming oblivious transfer and the hardness of factoring. Our protocol is highly modular, and its uppermost layer can be viewed as a template that generalizes the structure of prior works and leads to a simpler security proof. We introduce a combined sampling-and-sieving technique that eliminates both the inherent leakage in the approach of Frederiksen et al. (Crypto'18), and the dependence upon additively homomorphic encryption in the approach of Hazay et al. (JCrypt'19). We combine this technique with an efficient, privacy-free check to detect malicious behavior retroactively when a sampled candidate is not a biprime, and thereby overcome covert rejection-sampling attacks and achieve both asymptotic and concrete efficiency improvements over the previous state of the art.